Vertex-algebra cohomology conjecture for the smooth cone

Let $\mathcal{C}$ be the cone in the source, let $A_f$ denote the corresponding finite or smooth-part algebra, and let $H_{\mathfrak{a}}^{i+\frac{\infty}{2}}(A_f)$ be the semi-infinite local-cohomology groups. Vertex-algebra cohomology conjecture. The groups constructed from geometrically defined vertex operator algebras for the smooth part of the cone $\mathcal{C}$ coincide with

$$H_{\mathfrak{a}}^{i+\frac{\infty}{2}}(A_f).$$

This would identify the algebraic semi-infinite local-cohomology construction with the vertex-operator-algebra cohomology theories developed in the cited works.